Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10,13,12)(11,15,16,14)(17,18,20,22)(19,23,24,21);(1,2)(3,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23);(2,4)(18,22)(21,23);(6,8)(11,16)(14,15)(19,24)(21,23);(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=fi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239781, #72262085, #72262141, … |
| maximal normal subgroup of | #72239781, #72262085, #72262141, … |
| maximal quotient of | #72256869, #72298425, #72298465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 517119);